#484006

order	512
nilpotency class	4
isoclinism	isoclinic class of D₄×D₁₆
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#33666	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18,23,28,24,29,21,27)(19,30,33,40,34,47,32,46)(20,26,35,44,36,45,22,37)(25,39,42,48,43,38,41,31);(17,19)(18,25)(20,31)(21,32)(22,38)(23,33)(24,34)(26,40)(27,41)(28,42)(29,43)(30,37)(35,39)(36,48)(44,47)(45,46);(1,3,4,10,5,9,11,15)(2,6,7,13,8,12,14,16)(17,20,23,35,24,36,21,22)(18,26,28,44,29,45,27,37)(19,31,33,39,34,48,32,38)(25,40,42,47,43,46,41,30);(3,9)(6,12)(10,15)(13,16)(17,21,24,23)(18,27,29,28)(19,32,34,33)(20,22,36,35)(25,41,43,42)(26,37,45,44)(30,46,47,40)(31,38,48,39)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=i,e^2=gi,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eg,g^b=gi,d^c=dhi

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260372, #72260377, #72260406
maximal normal subgroup of	#72260372, #72260377, #72260406

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391202);